Suppose I have a sample $\textbf{x} = (x_1, \dots, x_n)$ from stationary and $\beta$-mixing time series. I want to estimate the scalar parameter $\theta$ of the distribution using MLE, i.e., $$ \hat{\theta}_{MLE} = \text{argmax}_{\theta} \;\mathbf{L}(\theta, \textbf{x}) $$
I am interested in the case when I will take not the full sample, but split it into two subsamples: $\textbf{x}_{1, n/2} = (x_1, \dots, x_{n/2})$ and $\textbf{x}_{n/2 + 1, n} = (x_{n/2 + 1}, \dots, x_n)$ and then I will calculate MLE's based on these two subsamples: $$ \hat{\theta}_{1, n/2} = \text{argmax}_{\theta} \;\mathbf{L}(\theta, \textbf{x}_{1, n/2}) $$ $$ \hat{\theta}_{n/2+1, n} = \text{argmax}_{\theta} \;\mathbf{L}(\theta, \textbf{x}_{n/2 + 1, n}) $$ Will these estimates be asymptotically independent as $n \to \infty$?
Would be grateful if you'd share any ideas.